Abstract. We give an expression for number of points for the family of Dwork K3 surfacesover finite fields of order q ≡ 1 (mod 4) in terms of Greene's finite field hypergeometric functions. We also develop hypergeometric point count formulas for all odd primes using McCarthy's p-adic hypergeometric function. Furthermore, we investigate the relationship between certain period integrals of these surfaces and the trace of Frobenius over finite fields. We extend this work to higher dimensional Dwork hypersurfaces.

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Abstract. We extend our previous work on hypergeometric point count formulas by proving that we can express the number of points on families of Dwork hypersurfacesin terms of Greene's finite field hypergeometric functions. We prove that when d is odd, the number of points can be expressed as a sum of hypergeometric functions plus (q d−1 − 1)/(q − 1) and conjecture that this is also true when d is even. The proof rests on a result that equates certain Gauss sum expressions with finite field hypergeometric functions. Furthermore, we discuss the types of hypergeometric terms that appear in the point count formula and give an explicit formula for Dwork threefolds.

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Abstract. Inspired by a result of Manin, we study the relationship between certain period integrals and the trace of Frobenius of genus 3 generalized Legendre curves. We show that both of these properties can be computed in terms of "matching" classical and finite field hypergeometric functions, a phenomenon that has also been observed in elliptic curves and many higher dimensional varieties.